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Theorem carden2b 9388
Description: If two sets are equinumerous, then they have equal cardinalities. (This assertion and carden2a 9387 are meant to replace carden 9965 in ZF without AC.) (Contributed by Mario Carneiro, 9-Jan-2013.) (Proof shortened by Mario Carneiro, 27-Apr-2015.)
Assertion
Ref Expression
carden2b (𝐴𝐵 → (card‘𝐴) = (card‘𝐵))

Proof of Theorem carden2b
StepHypRef Expression
1 cardne 9386 . . . . 5 ((card‘𝐵) ∈ (card‘𝐴) → ¬ (card‘𝐵) ≈ 𝐴)
2 ennum 9368 . . . . . . . 8 (𝐴𝐵 → (𝐴 ∈ dom card ↔ 𝐵 ∈ dom card))
32biimpa 479 . . . . . . 7 ((𝐴𝐵𝐴 ∈ dom card) → 𝐵 ∈ dom card)
4 cardid2 9374 . . . . . . 7 (𝐵 ∈ dom card → (card‘𝐵) ≈ 𝐵)
53, 4syl 17 . . . . . 6 ((𝐴𝐵𝐴 ∈ dom card) → (card‘𝐵) ≈ 𝐵)
6 ensym 8550 . . . . . . 7 (𝐴𝐵𝐵𝐴)
76adantr 483 . . . . . 6 ((𝐴𝐵𝐴 ∈ dom card) → 𝐵𝐴)
8 entr 8553 . . . . . 6 (((card‘𝐵) ≈ 𝐵𝐵𝐴) → (card‘𝐵) ≈ 𝐴)
95, 7, 8syl2anc 586 . . . . 5 ((𝐴𝐵𝐴 ∈ dom card) → (card‘𝐵) ≈ 𝐴)
101, 9nsyl3 140 . . . 4 ((𝐴𝐵𝐴 ∈ dom card) → ¬ (card‘𝐵) ∈ (card‘𝐴))
11 cardon 9365 . . . . 5 (card‘𝐴) ∈ On
12 cardon 9365 . . . . 5 (card‘𝐵) ∈ On
13 ontri1 6218 . . . . 5 (((card‘𝐴) ∈ On ∧ (card‘𝐵) ∈ On) → ((card‘𝐴) ⊆ (card‘𝐵) ↔ ¬ (card‘𝐵) ∈ (card‘𝐴)))
1411, 12, 13mp2an 690 . . . 4 ((card‘𝐴) ⊆ (card‘𝐵) ↔ ¬ (card‘𝐵) ∈ (card‘𝐴))
1510, 14sylibr 236 . . 3 ((𝐴𝐵𝐴 ∈ dom card) → (card‘𝐴) ⊆ (card‘𝐵))
16 cardne 9386 . . . . 5 ((card‘𝐴) ∈ (card‘𝐵) → ¬ (card‘𝐴) ≈ 𝐵)
17 cardid2 9374 . . . . . 6 (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴)
18 id 22 . . . . . 6 (𝐴𝐵𝐴𝐵)
19 entr 8553 . . . . . 6 (((card‘𝐴) ≈ 𝐴𝐴𝐵) → (card‘𝐴) ≈ 𝐵)
2017, 18, 19syl2anr 598 . . . . 5 ((𝐴𝐵𝐴 ∈ dom card) → (card‘𝐴) ≈ 𝐵)
2116, 20nsyl3 140 . . . 4 ((𝐴𝐵𝐴 ∈ dom card) → ¬ (card‘𝐴) ∈ (card‘𝐵))
22 ontri1 6218 . . . . 5 (((card‘𝐵) ∈ On ∧ (card‘𝐴) ∈ On) → ((card‘𝐵) ⊆ (card‘𝐴) ↔ ¬ (card‘𝐴) ∈ (card‘𝐵)))
2312, 11, 22mp2an 690 . . . 4 ((card‘𝐵) ⊆ (card‘𝐴) ↔ ¬ (card‘𝐴) ∈ (card‘𝐵))
2421, 23sylibr 236 . . 3 ((𝐴𝐵𝐴 ∈ dom card) → (card‘𝐵) ⊆ (card‘𝐴))
2515, 24eqssd 3982 . 2 ((𝐴𝐵𝐴 ∈ dom card) → (card‘𝐴) = (card‘𝐵))
26 ndmfv 6693 . . . 4 𝐴 ∈ dom card → (card‘𝐴) = ∅)
2726adantl 484 . . 3 ((𝐴𝐵 ∧ ¬ 𝐴 ∈ dom card) → (card‘𝐴) = ∅)
282notbid 320 . . . . 5 (𝐴𝐵 → (¬ 𝐴 ∈ dom card ↔ ¬ 𝐵 ∈ dom card))
2928biimpa 479 . . . 4 ((𝐴𝐵 ∧ ¬ 𝐴 ∈ dom card) → ¬ 𝐵 ∈ dom card)
30 ndmfv 6693 . . . 4 𝐵 ∈ dom card → (card‘𝐵) = ∅)
3129, 30syl 17 . . 3 ((𝐴𝐵 ∧ ¬ 𝐴 ∈ dom card) → (card‘𝐵) = ∅)
3227, 31eqtr4d 2857 . 2 ((𝐴𝐵 ∧ ¬ 𝐴 ∈ dom card) → (card‘𝐴) = (card‘𝐵))
3325, 32pm2.61dan 811 1 (𝐴𝐵 → (card‘𝐴) = (card‘𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398   = wceq 1530  wcel 2107  wss 3934  c0 4289   class class class wbr 5057  dom cdm 5548  Oncon0 6184  cfv 6348  cen 8498  cardccrd 9356
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-ord 6187  df-on 6188  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-er 8281  df-en 8502  df-card 9360
This theorem is referenced by:  card1  9389  carddom2  9398  cardennn  9404  cardsucinf  9405  pm54.43lem  9420  nnadju  9615  ficardun  9616  ackbij1lem5  9638  ackbij1lem8  9641  ackbij1lem9  9642  ackbij2lem2  9654  carden  9965  r1tskina  10196  cardfz  13330
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